A note on 3-connected hourglass-free claw-free Hamilton-connected graphs

Abstract

An hourglassΓ0 is the graph with degree sequence (4, 2, 2, 2, 2) and a claw is the bipartite complete graph K1,3. In this paper, we show that(1)every 3-connected, essentially 4-connected {K1,3,Γ0}-free graph is Hamilton-connected,(2)every 3-connected {K1,3,Γ0,P16}-free graph is Hamilton-connected, where (1) improves similar results of (Li et al. (2008) [11]) and hence (Broersma et al. (2001) [4]); (2) settles one conjecture posed in (Ryjáček et al. (2018) [18]).